NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

ecause one can quickly adjust the Q and R matrices to change the state penalty
weightings. Finally, this design technique can be interpreted as improving upon the
initial stabilizing control (developed through some sub-optimal approach, such as a
backstepping or PBC strategy) at each iteration of the algorithm.
3.5.2 The H1 Problem
The successive Galerkin approximation technique for the Hamilton-Jacobi-
Isaacs (HJI) equation is described in [5]. The basic idea is similar to the previous
section except that the successive iteration step requires two nested loops. The non-
linear H1 problem data is given by
_x = f(x)+g(x)u + k(x)w
Z T
0
l(x)+kuk 2
R dt
x(0) = 0 8 T 0
2
Z T
0
kwk 2
P dt
where it is also desirable to compute the smallest possible > 0. In other words, the
objective is to minimize the L 2 gain from an exogenous disturbance signal, w, to an
output de ned by R l(x)+kuk 2
R dt. The solution to this minimization problem is the
HJI equations, given by
@V T
@x f + hT h + 1
4
@V
@x
1
(
2 2 kP ;1 k T ; gR ;1 g T T
@V
)
@x
=0: (3.28)
In a manner directly analogous to the previous section we can write the HJI equation
in a way that decouples u, w, and V , and then we can solve for the optimal u
iteratively: we start, as before, with an initial stabilizing control u (0) , and then iterate
between the disturbance and V untilitistheworst disturbance for the given control.
Then we update the control u (1) and iteratively compute the worst disturbance for
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